; 1; Harvard University, Department of Mathematics; 2; MIT, Computer Science and Artificial Intelligence Laboratory

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ABSTRACT

Abstract. We study the problem of computing the k-th term of the Farey sequence of order n, for given n and k. Several methods for generating the entire Farey sequence are known. However, these algorithms require at least quadratic time, since the Farey sequence has Θ(n 2) elements. For the problem of finding the k-th element, we obtain an algorithm that runs in time O(n lg n) and uses space O ( √ n). The same bounds hold for the problem of determining the rank in the Farey sequence of a given fraction. A more complicated solution can reduce the space to O(n 1/3 (lg lg n) 2/3), and, for the problem of determining the rank of a fraction, reduce the time to O(n). We also argue that an algorithm with running time O(poly(lg n)) is unlikely to exist, since that would give a polynomial-time algorithm for integer factorization. 1